The Pestov–Ionin theorem in the differential geometry of plane curves states that every simple closed curve of curvature at most one encloses a unit disk.

History and generalizations

Although a version of this was published for convex curves by Wilhelm Blaschke in 1916, ^[1] it is named for German Gavrilovich Pestov [ ru] and Vladimir Kuzmich Ionin [ ru], who published a version of this theorem in 1959 for non-convex doubly differentiable (${\displaystyle C^{2}}$) curves, the curves for which the curvature is well-defined at every point. ^[2] The theorem has been generalized further, to curves of bounded average curvature (singly differentiable, and satisfying a Lipschitz condition on the derivative), ^[3] and to curves of bounded convex curvature (each point of the curve touches a unit disk that, within some small neighborhood of the point, remains interior to the curve). ^[4]

Applications

The theorem has been applied in algorithms for motion planning. In particular it has been used for finding Dubins paths, shortest routes for vehicles that can move only in a forwards direction and that can turn left or right with a bounded turning radius. ^{[3] [5]} It has also been used for planning the motion of the cutter in a milling machine for pocket machining, ^[4] and in reconstructing curves from scattered data points. ^[6]

References